Let \(M\)  be a stable matching in sr. For any two agents \(a_i\)  and \(a_j\) , we denote by \(\text{rank}(a_i,a_j)\)  the position of \(a_j\)  in \(a_i\) 's preference list and define the rank of \(a_i\)  with respect to matching \(M\)  as \(\text{rank}(a_i,M(a_i))\) . The degree of \(M\)  is the largest rank of all agents in \(M\) . A stable matching \(M\)  is minimum regret if the degree of \(M\)  is minimum over all stable matchings. [1]} describe an algorithm to find a minimum regret stable matching, in a solvable instance \(I\)  of sr, in \(O(n^2)\)  time. As in the smi case, the profile of \(M\)  is given by the vector \(p(M) = \langle p_1, p_2, ..., p_n \rangle \)  where
\(p_k = |\lbrace a_i\in A : \text{rank}(a_i,M(a_i))=k\rbrace |\)  for each \(k\)  \((1 \le k \le n)\) . A stable matching \(M\)  is then rank-maximal if \(p(M)\)  is lexicographically maximum, taken over all stable matchings. Finally, a stable matching \(M\)  is generous if the reverse profile \(p_r(M) = \langle p_n, p_{n-1}, ..., p_1 \rangle \)  is lexicographically minimum, taken over all stable matchings.
